Properties

Label 1200.3.bg
Level $1200$
Weight $3$
Character orbit 1200.bg
Rep. character $\chi_{1200}(193,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $72$
Newform subspaces $17$
Sturm bound $720$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 17 \)
Sturm bound: \(720\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1200, [\chi])\).

Total New Old
Modular forms 1032 72 960
Cusp forms 888 72 816
Eisenstein series 144 0 144

Trace form

\( 72q + O(q^{10}) \) \( 72q - 24q^{13} + 8q^{17} - 96q^{23} - 128q^{31} - 48q^{33} - 8q^{37} + 64q^{41} + 64q^{43} + 192q^{47} + 192q^{51} + 56q^{53} - 64q^{61} - 480q^{67} - 512q^{71} + 232q^{73} - 648q^{81} + 544q^{83} + 288q^{87} + 576q^{91} - 96q^{93} - 56q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1200.3.bg.a \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-24\) \(q-\beta _{1}q^{3}+(-6+6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.b \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-16\) \(q+\beta _{1}q^{3}+(-4+4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.c \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-16\) \(q-\beta _{1}q^{3}+(-4+4\beta _{2}+3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.d \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-16\) \(q-\beta _{3}q^{3}+(-4+4\beta _{1}-4\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots\)
1200.3.bg.e \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-12\) \(q+\beta _{1}q^{3}+(-3+3\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.f \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{1}q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.g \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+5\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.h \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+2\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.i \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{3}+3\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.j \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{3}+6\beta _{1}q^{7}-3\beta _{2}q^{9}+18q^{11}+\cdots\)
1200.3.bg.k \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{3}q^{3}+(1+2\beta _{1}+\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots\)
1200.3.bg.l \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{1}q^{3}+(2-2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.m \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(16\) \(q+\beta _{1}q^{3}+(4-4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.n \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(16\) \(q+\beta _{1}q^{3}+(4-4\beta _{2}-3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.o \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(20\) \(q-\beta _{1}q^{3}+(5-5\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.p \(4\) \(32.698\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(24\) \(q-\beta _{1}q^{3}+(6-6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.q \(8\) \(32.698\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{3}+(\beta _{3}-\beta _{6})q^{7}-3\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)